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Divergences Test Statistics for Discretely Observed Diffusion Processes
Alessandro De Gregorio, Università di Milano, Italy
Stefano Iacus, Department of Economics, Business and Statistics, University of Milan, IT
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ABSTRACT:
In this paper we propose the use of $\phi$-divergences as test
statistics to verify simple hypotheses about a one-dimensional
parametric diffusion process $\de X_t = b(X_t, \theta)\de t +
\sigma(X_t, \theta)\de W_t$, from discrete observations
$\{X_{t_i}, i=0, \ldots, n\}$ with $t_i = i\Delta_n$, $i=0, 1,
\ldots, n$, under the asymptotic scheme $\Delta_n\to0$,
$n\Delta_n\to\infty$ and $n\Delta_n^2\to 0$. The class of
$\phi$-divergences is wide and includes several special members
like Kullback-Leibler, R\'enyi, power and $\alpha$-divergences. We
derive the asymptotic distribution of the test statistics based on
$\phi$-divergences. The limiting law takes different forms
depending on the regularity of $\phi$. These convergence differ
from the classical results for independent and identically
distributed random variables. Numerical analysis is used to show
the small sample properties of the test statistics in terms of
estimated level and power of the test.
SUGGESTED CITATION:
Alessandro De Gregorio and Stefano Iacus,
"Divergences Test Statistics for Discretely Observed Diffusion Processes"
(August 2008).
UNIMI - Research Papers in Economics, Business, and Statistics.
Statistics and Mathematics.
Working Paper 38.
http://services.bepress.com/unimi/statistics/art38
Paper presented by C. Tommasi.
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